See also: Welcome to postcarding!; More postcarding resources
A good design has at least a short message, and you may want to accentuate with clip art or a (properly licensed and content-appropriate) stock photo.
Some designs aren't appropriate for some campaigns (e.g., because of being partisan), and that's OK. A design that's inappropriate for one campaign may be the ideal design for another.
If seeing existing designs might help inspire you, Postcards to Voters has some (note that most of theirs are partisan), and here are mine.
If your design includes a bleed (see getting postcards printed at a local print shop for more info on that), you'll want to include crop marks at each of the corners of the design so the print shop knows where to cut the paper. You can see those at the corners of this image (note: don't use this image for printing; get the PDF from my designs):
I made this a few years ago for folks who want to print any 4-by-6 image as a custom postcard. Print your image on the front, and this on the back.
This design is available in 1-up (6-by-4-inch), 3-up (6-by-12-inch), and 6-up (12-by-12-inch) layouts.
Here's how I print my own postcards at home from my postcard designs.
The most efficient way is with 12-by-12-inch cardstock, because each sheet evenly divides into six 4-by-6-inch postcards. You can cut slightly smaller postcards from letter-size cardstock, but you'll have scraps and can only do 3 cards per sheet.
(Note that “postcard” refers to a specified thing, and that specification includes a finite range of sizes. 4-by-6 is within that range. 5-by-7 is too large; USPS will consider it a letter. All of my postcarding information is designed for 4-by-6.)
Unless you have a wide-format printer that can print directly onto 12-inch sheets, your printer probably tops out at 11 inches in width. This means you'll need to cut the sheets down to fit. I use a guillotine to cut them down to 6 by 12, and then print one column of postcards on each strip.
My printer can do two-sided printing of these when using AirPrint, so I typically do that: I use Preview on the Mac to combine the 3-up front and the 3-up back into one two-page PDF, and print however many copies of that. (That's how I made the two-page PDFs in the list above.)
Remember the math involved: each 12-by-12 sheet divides into two 6-by-12 strips, and each strip holds three postcards. That means however many sheets you pull out of the pack, multiply by 6 to determine the number of postcards you'll print. Or, the other way around, if you know you'll need to print a certain number of postcards (e.g., because that's how many addresses you got), divide by 6 to get the number of sheets to pull out of the pack.
Once the strips are printed on both sides, I load them back into the guillotine to split them into thirds. I set the stop at 8 inches, and cut one postcard off of every strip; then move the stop to 4 inches, and divide each strip into the two remaining cards.
I do have a Silhouette cutting machine, but sticking two strips onto the cutting mat and then peeling them off again is more work than it's worth for, effectively, two cuts. If I had a wide-format printer, then using the cutting machine might make more sense for cutting the whole 12-by-12 sheets into six cards.
If all you have is a “photo printer” that does sizes like 4-by-6 but not 6-by-anything, then you'll need to cut the cardstock all the way down to 4-by-6 to start with, or buy blank (unruled) 4-by-6 index cards, and feed those directly into a printer capable of printing on such sizes. (You might not be able to do duplex printing—that is, you might have to print all one side and then all the other. Pay attention to which side gets printed on! Some printers print on the upper side, some on the lower; there should be a diagram on the input tray to tell you which.) Also, you will pay a mint in ink, so that's the point at which it might be cheaper to either get your cards professionally printed or buy off-the-shelf designs.